The central limit theorem for weighted empirical processes indexed by sets

Sufficient conditions are found for the weak convergence of a weighted empirical process {(ν_n(C)/q(P(C))) 1 _{[P(C) λn]}: C }, indexed by a class of sets and weighted by a function q of the size of each set. We find those functions q which allow weak convergence to a sample-continuous Gaussian process, and, given q, determine the fastest rate at which one may allow λ_n → 0.     

A limit in probability in a W-algebra is unique

Let be the space of self-adjoint Segal measurable operators affiliated to a W^*-algebra (for x=∫λe_x(dλ), x e_x((−∞, −n)(n, ∞)) is a finite projection in for n large enough). The limit in probability is unique in . x_n → x in probability e_nProj ;e_n→1 strongly and (x_n − x) e_n → 0. The following proposition proves to be important in the investigation of . If 1 − ƒ is a finite projection in and 1 − e_n → 0 strongly, e_nProj , then strongly.     

Maximum likelihood estimation of a change-point in the distribution of independent random variables: General multiparameter case

In a sequence ofn independent random variables the pdf changes fromf(x, 0) tof(x, 0 + δv_n⁻¹) after the firstnλ variables. The problem is to estimateλ (0, 1 ), where 0 and δ are unknownd-dim parameters andv_n → ∞ slower thann^1/2. Let_n denote the maximum likelihood estimator (mle) ofλ. Analyzing the local behavior of the likelihood function near the true parameter values it is shown under regularity conditions that ifn_n²(− λ) is bounded in probability asn → ∞, then it converges in law to the timeT_(δjδ)^1/2 at which a two-sided Brownian motion (B.M.) with drift1/2(δ′Jδ)^1/2ton(−∞, ∞) attains its a.s. unique minimum, whereJ denotes the Fisher-information matrix. This generalizes the result for small change in mean of univariate normal random variables obtained by Bhattacharya and Brockwell (1976,Z. Warsch. Verw. Gebiete37, 51–75) who also derived the distribution ofT_μ forμ > 0. For the general case an alternative estimator is constructed by a three-step procedure which is shown to have the above asymptotic distribution. In the important case of multiparameter exponential families, the construction of this estimator is considerably simplified.     

Strong limit theorems for quasi-orthogonal random fields

This is a systematic and unified treatment of a variety of seemingly different strong limit problems. The main emphasis is laid on the study of the a.s. behavior of the rectangular means ζ_mn = 1/(λ₁(m) λ₂(n)) Σ_i=1^m Σ_k=1ⁿ X_ik as either max{m, n} → ∞ or min{m, n} → ∞. Here {X_ik: i, k ≥ 1} is an orthogonal or merely quasi-orthogonal random field, whereas {λ₁(m): m ≥ 1} and {λ₂(n): n ≥ 1} are nondecreasing sequences of positive numbers subject to certain growth conditions. The method applied provides the rate of convergence, as well. The sufficient conditions obtained are shown to be the best possible in general. Results on double subsequences and 1-parameter limit theorems are also included.     

q-Bernstein polynomials and their iterates

Let B_n( f,q;x), n=1,2,… be q-Bernstein polynomials of a function f : [0,1]→C. The polynomials B_n( f,1;x) are classical Bernstein polynomials. For q≠1 the properties of q-Bernstein polynomials differ essentially from those in the classical case. This paper deals with approximating properties of q-Bernstein polynomials in the case q>1 with respect to both n and q. Some estimates on the rate of convergence are given. In particular, it is proved that for a function f analytic in {z: |z|<q+} the rate of convergence of {B_n( f,q;x)} to f(x) in the norm of C[0,1] has the order q⁻ⁿ (versus 1/n for the classical Bernstein polynomials). Also iterates of q-Bernstein polynomials {B_n^j_n( f,q;x)}, where both n→∞ and j_n→∞, are studied. It is shown that for q(0,1) the asymptotic behavior of such iterates is quite different from the classical case. In particular, the limit does not depend on the rate of j_n→∞.     

Two-pattern strings II—frequency of occurrence and substring complexity

The previous paper in this series introduced a class of infinite binary strings, called two-pattern strings, that constitute a significant generalization of, and include, the much-studied Sturmian strings. The class of two-pattern strings is a union of a sequence of increasing (with respect to inclusion) subclasses T_λ of two-pattern strings of scope λ, λ=1,2,…. Prefixes of two-pattern strings are interesting from the algorithmic point of view (their recognition, generation, and computation of repetitions and near-repetitions) and since they include prefixes of the Fibonacci and the Sturmian strings, they merit investigation of how many finite two-pattern strings of a given size there are among all binary strings of the same length. In this paper we first consider the frequency f_λ(n) of occurrence of two-pattern strings of length n and scope λ among all strings of length n on {a,b}: we show that lim_n→∞f_λ(n)=0, but that for strings of lengths n2λ, two-pattern strings of scope λ constitute more than one-quarter of all strings. Since the class of Sturmian strings is a subset of two-pattern strings of scope 1, it was natural to focus the study of the substring complexity of two-pattern strings to those of scope 1. Though preserving the aperiodicity of the Sturmian strings, the generalization to two-pattern strings greatly relaxes the constrained substring complexity (the number of distinct substrings of the same length) of the Sturmian strings. We derive upper and lower bounds on C₁(k) (the number of distinct substring of length k) of two-pattern strings of scope 1, and we show that it can be considerably greater than that of a Sturmian string. In fact, we describe circumstances in which lim_k→∞(C₁(k)−k)=∞.     

On the Maximal Multiplicity of Parts in a Random Integer Partition

We study the asymptotic behavior of the maximal multiplicity μ_n = μ_n(λ) of the parts in a partition λ of the positive integer n, assuming that λ is chosen uniformly at random from the set of all such partitions. We prove that πμ_n/(6n)^1/2 converges weakly to max _jX_j/j as n→∞, where X₁, X₂, … are independent and exponentially distributed random variables with common mean equal to 1.2000 Mathematics Subject Classification: Primary—05A17; Secondary—11P82, 60C05, 60F05     

Sphericity test in a GMANOVA–MANOVA model with normal error

For the GMANOVA–MANOVA model with normal error: , , we study in this paper the sphericity hypothesis test problem with respect to covariance matrix: Σ=λI_q (λ is unknown). It is shown that, as a function of the likelihood ratio statistic Λ, the null distribution of Λ^2/n can be expressed by Meijer’s function, and the asymptotic null distribution of −2logΛ is (as n→∞). In addition, the Bartlett type correction −2ρlogΛ for logΛ is indicated to be asymptotically distributed as with order n⁻² for an appropriate Bartlett adjustment factor −2ρ under null hypothesis.     

The Lebesgue Constant for Higher Order Hermite-Fejér Interpolation on the Chebyshev Nodes

For a fixed integer m ≥ 0, and for n = 1, 2, 3, ..., let λ_{2m, n}(x) denote the Lebesgue function associated with (0, 1,..., 2m) Hermite-Fejér polynomial interpolation at the Chebyshev nodes {cos[(2k−1) π/(2n)]: k=1, 2, ..., n}. We examine the Lebesgue constant Λ_{2m, n} max{λ_{2m, n}(x): −1 ≤ x ≤ 1}, and show that Λ_{2m, n} = λ_{m, n}(1), thereby generalising a result of H. Ehlich and K. Zeller for Lagrange interpolation on the Chebyshev nodes. As well, the infinite term in the asymptotic expansion of Λ_{2m, n)} as n → ∞ is obtained, and this result is extended to give a complete asymptotic expansion for Λ_{2, n}.     

Large deviation for the empirical eigenvalue density of truncated Haar unitary matrices

Let U_m be an m×m Haar unitary matrix and U_[m,n] be its n×n truncation. In this paper the large deviation is proven for the empirical eigenvalue density of U_[m,n] as m/n→λ and n→∞. The rate function and the limit distribution are given explicitly. U_[m,n] is the random matrix model of quq, where u is a Haar unitary in a finite von Neumann algebra, q is a certain projection and they are free. The limit distribution coincides with the Brown measure of the operator quq.     

On the generalized Euler-Frobenius polynomial

In this paper, the properties of the generalized Euler-Frobenius polynomial Π_n(·, q) are studied. It is proved that its zeroes are separated by the factor q and their asymptotic behavior, as q → ∞, is given. As a consequence, it is shown that least squares spline approximation on a biinfinite geometric mesh can be bounded independently of the (local) mesh ratio q and that the norm of the inverse of the corresponding order kB-spline Gram matrix decreases monotonically to 2k − 1 for large q, as q → ∞.     

On the Convergence and Iterates of q-Bernstein Polynomials

The convergence properties of q-Bernstein polynomials are investigated. When q1 is fixed the generalized Bernstein polynomials _nf of f, a one parameter family of Bernstein polynomials, converge to f as n→∞ if f is a polynomial. It is proved that, if the parameter 0<q<1 is fixed, then _nf→f if and only if f is linear. The iterates of _nf are also considered. It is shown that _n^Mf converges to the linear interpolating polynomial for f at the endpoints of [0,1], for any fixed q>0, as the number of iterates M→∞. Moreover, the iterates of the Boolean sum of _nf converge to the interpolating polynomial for f at n+1 geometrically spaced nodes on [0,1].     

Rate of Convergence of Positive Series

We investigate the rate of convergence of series of the form

Perturbation theory in large order

For many quantum mechanical models, the behavior of perturbation theory in large order is strikingly simple. For example, in the quantum anharmonic oscillator, which is defined by−″ + (x²/4+εx⁴/4−E)y=0, y(±∞)=0,the perturbation coefficients A_n in the expansion for the ground-state energysimplify dramatically as n → ∞:.We use the methods of applied mathematics to investigate the nature of perturbation theory in quantum mechanics and we show that its large-order behavior is determined by the semiclassical content of the theory. In quantum field theory the perturbation coefficients are computed by summing Feynman graphs. We present a statistical procedure in a simple λ⁴ model for summing the set of all graphs as the number of vertices → ∞. Finally, we discuss the connection between the large-order behavior of perturbation theory in quantum electrodynamics and the value of α, the charge on the electron.     

Polynomials of binomial type and compound Poisson processes

Let q_nand s_n, n ε N, respectively, be a set of polynomials of binomial type and a Sheffer set related to it, both having positive coefficients. Then q_n(x), x > 0 is connected with the probability that a compound Poisson process starting at zero is in state n at time τx andq_n(x)/q_n(1) is the probability generating function of the number of jumps of this process in [0, τ] given that it is in state n at time τ. The s_n admit similar interpretations when the initial distribution of the compound Poisson process is not concentrated at zero. The possible limits for n → ∞ ofq_n(x)/q_n(1)ands_n(x)/s_n(1) are studied.     

Iterative approximation of solutions to nonlinear equations involving m-accretive operators in Banach spaces

Let E be an arbitrary real Banach space and T :E→E be a Lipschitz continuous accretive operator. Under the lack of the assumption lim_n→∞α_n=lim_n→∞β_n=0, we prove that the Ishikawa iterative sequence with errors converges strongly to the unique solution of the equation x+Tx=f. Moreover, this result provides a convergence rate estimate for some special cases of such a sequence. Utilizing this result, we imply that if T :E→E is a Lipschitz continuous strongly accretive operator then the Ishikawa iterative sequence with errors converges strongly to the unique solution of the equation Tx=f. Our results improve, generalize and unify the ones of Liu, Chidume and Osilike, and to some extent, of Reich.     

On the completeness of the system of function {e sin λnx}n = 1

In this paper, the biorthogonal system corresponding to the system {e^−αnx sin nx}_{n = 1}^∞ is represented in an appropriate form so that it is possible to obtain sufficiently good estimates of its norm. Then, by the stability of a completeness property we prove that the system of functions {e^−αλ_nx sin λ_nx}_{n = 1}^∞ is complete.     

A Remez-Type Inequality for Non-dense Müntz Spaces with Explicit Bound

LetΛ :=(λ_k)^∞_k=0be a sequence of distinct nonnegative real numbers withλ₀ :=0 and ∑^∞_k=1 1/λ_k<∞. Let(0, 1) and(0, 1−) be fixed. An earlier work of the authors shows that [formula]is finite. In this paper an explicit upper bound forC(Λ, , ) is given. In the special caseλ_k :=k^α,α>1, our bounds are essentially sharp.     

Continuity in G

For a discrete group G, we consider βG, the Stone– ech compactification of G, as a right topological semigroup, and G^*=βGG as a subsemigroup of βG. We study the mappings λ_p^* :G^*→G^*and μ^* :G^*→G^*, the restrictions to G^* of the mappings λ_p :βG→βG and μ :βG→βG, defined by the rules λ_p(q)=pq, μ(q)=qq. Under some assumptions, we prove that the continuity of λ_p^* or μ^* at some point of G^* implies the existence of a P-point in ω^*.     

Derangement Characters of the Finite General Linear Group

We focus on derangement characters of GL(n,q) which depend solely on the dimension of the space of fixed vectors. This family includes Thoma characters which become asymptotically irreducible as n→∞. We find explicit decomposition of Thoma characters into irreducibles, construct further derangement characters and seek for extremes in the family derangement characters. Presented by A. VerschorenMathematics Subject Classification (2000) 20C15.